A valid categorical syllogism must conform to certain rules. These rules of syllogism

are the norms or standard that helps us to test the validity or the invalidity of the

moods. If we draw the conclusion in accordance with the rules of syllogism, the

argument is valid or else it becomes invalid.

The violation of any of the rule leads to a logical mistake otherwise called a logical

fallacy. Let us discuss the rules of syllogism and the corresponding fallacies that are

committed when the rules are violated. Mainly, we will deal with the following topics

while discussing the rules of syllogism. These are

(A) General Syllogistic Rules.

(B) Special Syllogistic Rules.

(C) Aristotle's Dictum.

(A) General Syllogistic Rules:

General Syllogistics rules are the fundamental and basic rules applicable to all

syllogisms in general. These are ten in number. Out of these ten, some are based on

the very definition of syllogism and some rules are derivative in nature. Let us discuss

them in detail.

Rule -1

Every syllogism must have three and only three terms neither more nor less. This rule

can not be regarded as a rule in the strict sense of the term because the very definition

of syllogism states that a syllogism must have three propositions and three terms.

These terms include the minor term, major term and the middle term. The middle term

keeps relationship with the extremes so that a conclusion is drawn. Similarly, we

cannot avoid either the major term or the minor term. Thus, in a syllogism, it is

necessary to have three and only terms.

If an argument has less than three terms (i.e. two terms), we cannot call it a syllogism,

rather it is a case of immediate inference.

For example,

All crocodiles are reptiles

Therefore, some reptiles are crocodiles

Here there are two terms and it is a case of immediate inference.

If an argument contains more than three terms (i.e. four terms), it cannot be called a

syllogism. We commit the fallacy of four terms. For example,

All cows are quadruped animals.

All dogs are faithful animals.

From this, we cannot draw any conclusion. It is a case of fallacy of four terms.

Sometimes a term is used in different senses in the same argument. In such a case, we

commit the Fallacy of Equivocation. This fallacy has three forms. When the major term

is used ambiguously, we call it the Fallacy of ambiguous major. For example,

Light is essential to guide our steps

Lead is not essential to guide our steps

Therefore, lead is not light.

The major term 'light' in the above argument has been used in one sense in the major

premise, but in another sense in the conclusion.

Similarly, when the minor term is used ambiguously, we commit the fallacy of

ambiguous minor.

For example,

No man is made of paper.

All pages are men.

Therefore, no pages are made of paper.

In this argument, the minor term 'page' has been used in two different senses.

When the middle term is used ambiguously, we commit the fallacy of ambiguous

middle. For example,

Sound travels at the rate of 1120 feet per second.

His knowledge of mathematics is sound

Therefore, his knowledge of mathematics travels at the rate of 1120 feet per second.

RuIe-2:

Every syllogism must have three and only three propositions. This is also not a rule in

the strict sense of the term. Like Rule-I, it states a necessary condition that a syllogism

must have three propositions out of which two are called premises and what follows

from the premises is called the conclusion. If we take less than three propositions, the

argument might become an immediate inference or if we take more than three

propositions, we get a train of syllogisms or Sorties.

Rule-3:

In a valid syllogism, the middle term must be distributed at least in one of the

premises.

The role of middle term in a syllogism is important because it connects both the

extremes. In order to establish a relation between the extremes (major and minor

terms) in the conclusion, extremes should be shown to be connected in some common

part of the middle term. In other

Words, for establishing a connection between the major and minor term in the

conclusion, at least one of them must be related to the whole of the middle term,

otherwise each of them might be connected only to with a different part of the middle

term. If the middle term is not distributed at least once in the premises, both the

extremes are not shown to be connected and we commit the fallacy of undistributed

middle. For example,

All dogs are quadruped.

All cats are quadruped.

So, all cats are dogs.

In both the premises, the middle term is undistributed (since A proposition doesn't

distribute its predicate). No conclusion is possible as the middle term is not properly

connected with the extremes. When this rule is violated we commit the, fallacy of

undistributed middle.

Rule-4

In a categorical syllogism, if a term is di

stributed in the conclusion, it must be distributed in the premise.

This rule states a necessary condition of deductive validity. The conclusion of a valid

deductive argument cannot be more general than the premises; the conclusion cannot

go beyond the premises. The conclusion can only make explicit what is implicitly

present in the premises. Syllogistic arguments, being deductive, must abide by this

condition.

The conclusion of a syllogism has two terms. These are minor term and major term.

Neither the major term nor the minor term should be distributed in the conclusion if it

is not distributed in the premise. Of course, the reverse is not a fallacy. A term which is

distributed in the premise may remain undistributed in the conclusion.

If the minor term is distributed in the conclusion but not distributed in the minor

premise, we commit the fallacy of illicit minor. For example,

AH men are rational.

All men are biped.

Therefore, all bipeds are rational.

Here the minor term 'biped' (subject term of conclusion) is distributed which is not

distributed in the minor premise (being the predicate of A proposition). So the fallacy

committed in this argument is illicit minor.

Similarly, if the major term is distributed in the conclusion without being distributed in

the major premise, we commit the fallacy of illicit major. For example,

All cows are quadruped.

No goats are cows.

Therefore, no goats are quadruped.

Here, the major term is distributed in the conclusion but not in the major premise

(since it is the predicate of an A proposition). So the fallacy of illicit major is committed

in this argument.

Rule-5

In a categorical syllogism, no conclusion can be obtained from two negative premises.

A negative proposition is one in which the predicate is denied of the subject i.e. the

predicate is negatively related with the subject. If both the premises are negative, the

middle term will be negatively related to the extremes and no relation can be

established between them. So a valid conclusion cannot be drawn. If we draw a

conclusion from two negative premises, we commit the fallacy of two negative

premises or fallacy of exclusive premises.

No artists are rich persons.

Some rich persons are not theists.

Therefore, some theists are not artists.

Since both the premises are negative, the conclusion (some theists are not artists) is

not valid and we commit the fallacy of two negative premises or fallacy of exclusive

premises.

Rule-6

In a categorical-syllogism, if either premise is negative, the conclusion must be

negative. According to Rule-5 stated above, we cannot draw any valid conclusion from

two negative premises. So, if one premise is negative, the other premise must be

affirmative. If one premise is affirmative and the other premise is negative, then a

relation of inclusion will be asserted between the middle term and one of the extremes

in the affirmative premise and the relation of exclusion will be asserted between the

middle term and the other extreme.

Thus, if one extreme is included in the middle term and the other excluded then there

can be the relation of exclusion between the extremes, and they cannot have

affirmative relation in the conclusion. Therefore, the conclusion will be negative. For

example,

No poets are scientists.

Some philosophers are poets.

Therefore, some philosophers are not scientists.

This conclusion (negative one) is a valid conclusion. But if we draw any affirmative

conclusion (such as "Some philosophers are scientists") from the above premises, it

would be a fallacious conclusion. Here, we would have committed the fallacy of

drawing an affirmative conclusion from a negative premise. Similarly, we can prove

that if the conclusion is negative, one of the premises must be negative

Rule- 7

In a categorical syllogism, if both the premises are affirmative, the conclusion must be

affirmative.

In an affirmative proposition, the predicate is affirmed of the subject. In other words,

in an affirmative premise a relation of inclusion is asserted. If both the premises are

affirmative, it is clear that the middle term is affirmatively connected with both the

extremes i.e. the minor term and the major term. Thus it is obvious that the minor term

and the major term are affirmatively related in the conclusion and the conclusion must

be an affirmative proposition.

Similarly, the converse also holds good. If the conclusion is affirmative, both the

premises must be affirmative.

Rule-8

In a categorical syllogism, if both the premises are particular, no conclusion follows. As

we know, there are two types of particular propositions. These are I and O

propositions. If both the premises are particular, then the possible combinations will

be I I, I O, O I and OO.

In 11 combination, since no term is distributed in I proposition the middle term is not

distributed. So this combination will not yield any conclusion (as per Rule 3) stated

above. In OO combination, there will be no conclusion (as per Rule 5) as it leads to the

fallacy of two negative premises.

Let us examine the combination of IO and O I. In either of the cases, since one premise

is negative, the conclusion will be negative. If the conclusion is negative, the predicate

of the conclusion (major term) will be distributed in the conclusion which could not be

distributed in the premise because there is only one term distributed in the premises

and it is reserved for the middle term to avoid the fallacy of undistributed middle).

So no conclusion follows from any of the combinations when both the premises are

particular. In other words, in a categorical syllogism at least one of the premises must

be universal.

RuIe-9

In a categorical syllogism, if one premise is particular, the conclusion will be

particular. If one premise is particular, the other premise will be universal because

according to Rule 8, stated above, from two particular premises no conclusion follows.

We have also seen that from two negative premises no conclusion can be drawn (See

Rule 5 stated above). So we get the following possible combinations.

AI, IA, AO, OA, EI, IE,

Let us examine each pair.

AI and I A:

In this combination, total number of terms distributed is one which is left for the

middle term (to avoid the fallacy of undistributed middle). So the conclusion will be a

proposition that does not distribute any term (to avoid the fallacy of either illicit major

or illicit minor). So the conclusion will be an I proposition which is particular.

AO and OA:

In this combination where one proposition is A and the other is O, the total number of

terms distributed in the premises is two, out of which one must be reserved for the

middle term to avoid the fallacy of undistributed middle and there is only one term left

as distributed. Since one premise is negative, the conclusion is bound to be negative

(as per Rule-6). Thus the conclusion will be a negative proposition and it will have only

one term distributed. The conclusion, therefore, must be an O proposition which is

particular.

EI and IE:

In this combination, total numbers of terms distributed in the premises are two out of

which one is reserved for the middle term. So there is only one term left as distributed

in the premise. Since one premise is E which is negative, the conclusion will be

negative where only one term can be distributed. So it must be an O proposition, which

is particular.

Thus we notice that if one premise is particular, the conclusion will be particular.

Rule-10:

In a categorical syllogism, if the major premise is particular and the minor premise is

negative then no conclusion follows.

If the minor premise is negative, the conclusion becomes negative (Rule 6) and the

major premise is bound to be affirmative (Rule 5). Thus the major premise is a

particular affirmative (T) proposition. Since the conclusion is negative its predicate

(major term) will be distributed in the conclusion which is not distributed in the major

premise. So the fallacy of illicit major will be committed.

Therefore, in a syllogism when the major premise is particular and minor premise is

negative, no conclusion can be drawn.